Session S49: Advances in Quantum Algorithms for Near-Term Applications

Enhancing initial state overlap through orbital optimization

Quantum phase estimation (PE) and its multiple variants require the efficient preparation of a state with high overlap with the targeted eigenstate. A few works were devoted to this task, notably introducing quantum circuits for preparing a configuration state function (CSF) instead of a single Slater determinant (SD) [2, 3, 4]. However, Lee et al. [1] showed that the overlap between a density matrix renormalization group approximation of the ground state and a single CSF (or SD) can be extremely small, thereby compromising the success of PE protocols. In addition, the overlap was shown to decrease exponentially on the studied iron-sulfur cluster molecules ranging from 2 to 8 metal centers including the sought-after FeMo cofactor. However, these single configuration states are defined in fixed orbital bases and orbital relaxation could improve their overlap with the ground state [1, 5]. Our work aims to show that this is indeed the case for classically demanding molecules. In this talk, I will introduce our comprehensive framework for orbital optimization and explore its implications for addressing the initial state overlap problem.

[1]  Seunghoon Lee, et al. Nature Communications, 14(1):1952, 2023.

[2]  Alessandro Carbone, et al. Symmetry, 14(3):624, 2022.

[3]  Kenji Sugisaki, et al. The Journal of Physical Chemistry A, 120(32):6459–6466, 2016.

[4]  Kenji Sugisaki, et al. Chemical Physics Letters, 737:100002, 2019.

[5] Norm M Tubman, et al. arXiv preprint arXiv:1809.05523, 2018.   

Tapered Quantum Phase Estimation

The quantum phase estimation (QPE) algorithm has been used in algorithmic settings to determine the phase of an input state, solve systems of linear equations, estimate quantum amplitudes, and for quantum principal component analysis. In standard QPE, ancilla qubits are prepared in an equal superposition state. In tapered quantum phase estimation (tQPE) we allow for an arbitrary initial ancilla state and optimize this state using modern signal analysis methods. In particular, given a number of ancilla qubits, we minimize the probability of outputting an estimate that deviates from the true phase by a certain amount in the worst-case and average error settings. Using frequency- (equivalently, phase-) concentrated tapered estimates, we show that the number of extra qubits in tQPE required to guarantee that the output of the algorithm is $delta$-close to the true phase with probability at least $1 - epsilon$ can be reduced asymptotically to $m =O(loglog(1/epsilon))$, which is optimal. We then show that the worst-case success probability for QPE (with no extra qubits, $m = 0$) can be improved by approximately $20\%$ using an optimal taper. Finally, we demonstrate that the mean success probability can be improved, relative to standard QPE, in all cases.   

Improving Phase Estimation by Post Processing QPE Circuit Output

Quantum Phase Estimation (QPE) is one of the most important and widely applied quantum computing algorithms known. Fundamental to Shor's algorithm, eigenvalue estimation, and other algorithms, improving the accuracy of phase estimation is central to the progression of quantum computing as a field. As quantum computers transition into the noisy intermediate-scale quantum (NISQ) era and eventually into more fault-tolerant implementations, the effective post-processing of QPE output will become increasingly pivotal to quantum computing applications. The classic method for evaluating QPE output is to take the "highest peaks" of the results for the estimate of the phases, leaving the other information behind. This method is accurate to one part in 2^(-n), where n is the number of qubits. In this talk, we present a software package that uses various numerical optimization methods along with machine learning to more accurately estimate the phases of unitary operators, reaching, especially for circuits with few qubits, well beyond the 2^(-n) limit of the traditional method. Using the techniques of the software package, a user can achieve the accuracy otherwise offered only by increasing the number of qubits of the QPE circuit or by doing iterative phase estimation, which requires mid-circuit measurement. Using this method allows the user to expand the capability of current QPE research, and we analyze the performance and applications of our method.   

Applications of projection algorithm for state preparation on quantum computers.

In this talk we will discuss applications of a recently developed algorithm for state preparation Phys. Rev. C 108, L031306 . 

After reviewing the algorithm and its scaling properties, we will discuss its application for state preparation for 2D Heisenberg Hamiltonians on a squared lattice and for some realistic nuclear physics Hamiltonians.

Given the high circuit depth for the systems considered we will show classical simulation results using a recently developed classical simulator  up to 30 qubits and tens of milions of gates https://doi.org/10.1145/3458817.3476169.   

Hidden time-evolution structure of the quantum approximate optimization algorithm when it is used for ground-state preparation

The quantum approximate optimization algorithm (QAOA) has emerged as an accurate and efficient approach to solve many optimization problems on quantum computers including ground-state preparation. The algorithm has many similarities to Trotterized time evolution. For example, one way to envision how it works is that it uses large inhomogenous time steps plus an additional variational principle to correct for the Trotter errors. By comparing optimized QAOA trajectories, we find it is closely related to local adiabatic time evolution. We then modify a Trotterized time evolution (based on the local adiabatic field) by adding in a variational scaling of the Hamiltonian at each time step. We achieve very high accuracy (fidelities of better than five 9s) with this approach. Our concrete example works with the transverse-field Ising model, but we believe the principles uncovered here are much more general and widely applicable.   

Angular momentum orbital space of molecules and extend systems for quantum computing

We developed highly efficient quantum algorithms for simulating the electronic Hamiltonian with spherical harmonics (SH) basis function by leveraging the angular momentum entanglement. For the evaluation of molecular integral using SH basis, our new scheme decomposes the computation into calculations depending solely on rotational property separated from the physical system. The coupling of angular momentum captures this sole dependency on the rotational properties. In quantum mechanics, the entanglement structure between the angular momentum states of an orbital and the surrounding field can be calculated using the Clebsch-Gordan (CG) transformation. The CG selection rule enforces the conservation of angular momentum by determining which angular momentum states are allowed in the resulting composite system. The SH basis is orthogonal and unitary, reducing the basis size needed to diagonalize the Coulomb operator. For the second quantized representation, original fermion ladder operators are projected to the spherical harmonic basis, forming the angular momentum orbital space. Our new approach greatly simplifies the direct simulation of this Hamiltonian using quantum computers with atomic orbital angular momentum block encoding. The encoding speedup for the small molecule is about six orders of magnitude compared to the existing method using Pauli operators.   

Continuous-Space Quantum Simulation: A Discretization-Free Approach with Hybrid Quantum-Classical Ansatze

Most quantum many-body systems including those of electronic structure and materials are natively described in first quantization. However, simulating continuous-space systems on quantum computers is challenging as the systems have to be discretized and mapped onto qubits while respecting the underlying exchange statistics. The discretization usually amounts to either discretizing space itself or choosing a suitable finite basis set which introduces errors and scales poorly with the number of particles. In this talk, I propose an alternative approach of harnessing quantum resources within Variational Monte Carlo simulations for continuous space problems which does not require any form of discretization. To that end, the wavefunction ansatz is partly represented by a parameterized quantum circuit and optimized similarly to the Neural Quantum States framework. I apply our hybrid quantum-classical algorithm to the paradigmatic 1d quantum rotor model and show how the accuracy of the ground state energy can be controlled by the circuit depth. The results are compared to typical classical ansatze such as Jastrow wavefunctions and MPS calculations on a discretized approximation of the model. In terms of the latter, I demonstrate a reduction in the number of qubits when simulating the system using our continuous space formalism. Finally, I show how this framework can be applied to fermionic systems in first quantization.   

Fault-tolerant quantum computation of molecular forces

The computation of observables beyond the ground state energy, such as dipole moments and molecular forces, is essential for many practical applications in drug design, including Molecular Dynamics. In this presentation, I will introduce a fault-tolerant quantum algorithm for calculating expectation values of molecular observables. Additionally, I will provide resource estimates in terms of the number of qubits and Toffoli gates required for the computation of nuclear forces in small molecules. In conclusion, I will put these initial resource estimates into perspective, discuss their practical implications for Molecular Dynamics on fault-tolerant quantum computers, and consider the steps needed to make Molecular Dynamics an applicable task on future fault-tolerant quantum computers.   

Challenges and opportunities for applying quantum computers to drug design

The current limitations of classical computing methods in accurately describing quantum systems hinder the application of quantum chemistry to drug design. More precise computations could replace many labor-intensive experiments. Quantum computations could offer key insights into chemical systems, justifying high computational costs in an industrial setting. However, to significantly impact the pharmaceutical industry, quantum computers must address a broader set of problems, including those involving large protein structures.  Significant advancements in hardware and quantum algorithms have reduced computational costs over the years, sparking optimism for the future use of quantum computing in quantum chemistry. However, harnessing the full potential of quantum computing in the pharmaceutical industry requires further improvements in hardware and novel algorithms. We will discuss these challenges and discuss several routes to achieve these goals and progress these challenges. Open research integrating academia and industry will help make quantum computing an essential tool for designing better drugs faster.

The current limitations of classical computing methods in accurately describing quantum systems hinder the application of quantum chemistry to drug design. More precise computations could replace many labor-intensive experiments, provided the computational cost is lower. Quantum computations could offer key insights into chemical systems, justifying high computational costs in an industrial setting. However, to significantly impact the pharmaceutical industry, quantum computers must address a broader set of problems, including those involving large protein structures. New methods that balance accuracy and time on quantum computers could be beneficial. Significant advancements in hardware and quantum algorithms have reduced computational costs over the years, sparking optimism for the future use of quantum computing in quantum chemistry. However, harnessing the full potential of quantum computing in the pharmaceutical industry requires further improvements in hardware and novel algorithms. We will discuss these challenges and explore several potential routes to achieve these goals.